# How do you find the limit of ln(lnt) as t->oo?

Jan 2, 2017

The answer is $\infty$

#### Explanation:

Since $\ln \left(t\right) \to \infty$ as $t \to \infty$, it follows that $\ln \left(\ln \left(t\right)\right) \to \infty$ as $t \to \infty$ (though very, very, slowly).

To illustrate how slowly $\ln \left(\ln \left(t\right)\right) \to \infty$ as $t \to \infty$, you might ask: how big should $t$ be so that $\ln \left(\ln \left(t\right)\right) > 10$? (for example)

To answer this, solve the inequality $\ln \left(\ln \left(t\right)\right) > 10$ by exponentiation: $\ln \left(\ln \left(t\right)\right) > 10 \iff \ln \left(t\right) > {e}^{10} \iff t > {e}^{{e}^{10}} \approx 9.4 \times {10}^{9565}$

In general, in order for $\ln \left(\ln \left(t\right)\right)$ to be bigger than an arbitrary $M > 0$, you need to choose $t$ so large that $t > {e}^{{e}^{M}}$.